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Chapter 0: Problem 56

Evaluate each exponential expression. $$\left(10 x^{2}\right)^{-3}$$

Short Answer

Expert verified
The simplified form of \(\left(10 x^{2}\right)^{-3}\) is \(1/(1000x^{6})\).

Step by step solution

01

Apply the Power of a Power Rule

In the expression \(\left(10 x^{2}\right)^{-3}\), apply the power of a power rule. This means multiplying the exponent inside the parentheses by the exponent outside: \(10^{-3} \cdot x^{-6}\).
02

Apply the Negative Exponent Rule

The negative exponent rule is now applied to both terms. This changes \(10^{-3}\) to \(1/10^{3}\) and \(x^{-6}\) to \(1/x^{6}\).
03

Simplify the Expression

Finally, the expression is simplified. This results in \(1/10^{3} \cdot 1/x^{6}\) being simplified to \(1/(1000x^{6})\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Power of a Power Rule
When working with exponential expressions, the power of a power rule is incredibly handy. This rule states that when you have an exponent raised to another exponent, you multiply the exponents. For example, in our expression \((10x^2)^{-3}\), you have the exponent 2 inside the parentheses and -3 outside. To simplify, multiply these exponents, resulting in \(10^{-3} \cdot x^{-6}\).

This step is crucial because it breaks down complex problems, making them easier to manage. The rule makes the calculations straightforward, especially when dealing with multiple variables or constants within an expression.
Negative Exponent Rule
The negative exponent rule is a fundamental part of understanding how to simplify expressions. A negative exponent indicates that the base should be taken as the reciprocal, turning negative exponents into positive ones.

In our example, \(10^{-3}\) becomes \(1/10^{3}\), and \(x^{-6}\) becomes \(1/x^{6}\). This conversion is essential because it shifts our perspective from complex to simplified forms, allowing for easier manipulation of expressions.
  • Negative exponents invert bases.
  • Always remember to convert to fractions when faced with negatives.
  • This transformation is universal for constants and variables alike.
Simplifying Expressions
Once you've applied the rules of exponents, simplifying the expression is the final step. This involves combining all parts into a single, cleaner expression.

After transforming exponents negatively, our expression \(1/10^{3} \cdot 1/x^{6}\) can be simplified to \(1/(1000x^{6})\). Simplification is about bringing the expression to its most elementary form, useful for reducing errors and making evaluations or further calculations easier.
  • Combine terms logically.
  • Simplifying reduces clutter.
  • Keep processes clear to avoid mistakes.
Keeping these points in mind ensures that complex expressions become manageable, providing clarity and confidence in problem-solving.

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Most popular questions from this chapter

Factor completely. $$-x^{2}-4 x+5$$

Solve each equation. $$\frac{1}{x^{2}-3 x+2}=\frac{1}{x+2}+\frac{5}{x^{2}-4}$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$8^{-\frac{1}{3}}=-2$$

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. To earn an A in a course, you must have a final average of at least \(90 \% .\) On the first four examinations, you have grades of \(86 \%, 88 \%, 92 \%,\) and \(84 \% .\) If the final examination counts as two grades, what must you get on the final to earn an A in the course?

A company wants to increase the \(10 \%\) peroxide content of its product by adding pure peroxide (100\% peroxide). If \(x\) liters of pure peroxide are added to 500 liters of its \(10 \%\) solution, the concentration, \(C,\) of the new mixture is given by $$C=\frac{x+0.1(500)}{x+500}$$ How many liters of pure peroxide should be added to produce a new product that is \(28 \%\) peroxide?
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